Duplicate and slide edge (rip from boundary)
Select the edge on the border and add a loop cut there. Since usual Loop Cut And Slide tool (Ctrl+R) won't do in this case due to 5-pole, use Offset Edge Slide instead.



This way will create triangle after cutting and sliding along edges which compose 5-pole, small  cleanup is required there to merge one vertex to another.  

See Offset Edge Slide in manual.

As mentioned in comments by @Leander, Offset Edge Slide indeed has options you can enable while executing it. Like Loop Cut and Slide, it provides:


Even mode which matches the shape of the created edge loop to existing edge loop from one side;
Flipped option which controls which edge loop will be taken into account by Even mode;
Clamp which allows to toggle limiting to existing edge loops.